Question

How do you integrate `[ln(lnx)]/[x] dx`?

Answer 1

lnx ln ln x- ln x

Explanation:

This can be done by u substitution. Let lnx =u, so that `1/x dx =du`

Answer 2

`int(ln(lnx))/xdx=lnxln(lnx)-lnx+c`

Explanation:

Let `z=lnx` then `dz=dx/x`

Hence `int(ln(lnx))/xdx=intlnzdz`

Now using integration by parts, if `u=lnz` and `v=z`

As `intudv=uv-intvdu`, we have

`intlnzdz=lnzxxz-intzdz/z=zlnz-z`

Hence `int(ln(lnx))/xdx=lnxln(lnx)-lnx+c`